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On the global behavior of a high-order rational difference equation. (English) Zbl 1198.39023
Summary: We consider the $\left(k+1\right)$-order rational difference equation ${y}_{n+1}=\frac{p+q{y}_{n}+r{y}_{n-k}}{1+{y}_{n-k}}$ where $k\in \left\{1,2,3,\cdots \right\}$, and the initial conditions ${y}_{-k},\cdots ,{y}_{-1},{y}_{0}$ and the parameters $p,q$ and $r$ are non-negative. We investigate the global stability, the periodic character and the boundedness nature of solutions of the above mentioned difference equation. In particular, our results solve the open problem introduced by M. R. S. Kulenović and G. Ladas [Dynamics of second order rational difference equations. With open problems and conjectures. Boca Raton, FL: Chapman & Hall/CRC (2002; Zbl 0981.39011)].
##### MSC:
 39A23 Periodic solutions (difference equations) 39A30 Stability theory (difference equations) 39A22 Growth, boundedness, comparison of solutions (difference equations)
##### References:
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